Last.fm Artist Similarity Graph Analysis

Network Structure and Distribution Analysis

import gzip
import json
import pickle
import warnings
from pathlib import Path

import numpy as np
import pandas as pd
import plotly.graph_objects as go
from IPython.display import HTML, Markdown, display
from plotly.subplots import make_subplots
from scipy.stats import gaussian_kde

from plot_template import setup_custom_template

warnings.filterwarnings("ignore")

setup_custom_template()

# Load data
metrics_path = Path("metrics/graph_metrics.json")
dist_path = Path("metrics/distributions.pkl.gz")
sample_path = Path("metrics/distributions_sample.json")

# Load metrics
with metrics_path.open() as f:
    metrics = json.load(f)

# Load distributions (try pickle first, fall back to JSON)
try:
    with gzip.open(dist_path, "rb") as f:
        distributions = pickle.load(f)
except Exception:
    with sample_path.open() as f:
        distributions = json.load(f)

# Extract data - new structure has everything at top level
dataset_info = metrics.get("dataset_info", {})
basic_metrics = metrics.get("basic_metrics", {})
degree_stats = metrics.get("degree_stats", {})
weight_stats = metrics.get("weight_stats", {})
power_law_fits = metrics.get("power_law_fits", {})

Executive Summary

This report analyzes the structure of a large-scale music artist similarity graph derived from Last.fm data. The network exhibits scale-free properties with distinct patterns in connectivity and similarity distributions.

Graph Overview

# Create DataFrame for basic statistics
stats_data = {
    "Metric": [
        "Total Nodes",
        "Total Edges",
        "Source Nodes",
        "Graph Density",
        "Reciprocity",
        "Average Degree",
    ],
    "Value": [
        f"{dataset_info['nodes']:,}",
        f"{dataset_info['edges']:,}",
        f"{dataset_info['source_nodes']:,}",
        f"{basic_metrics['density']:.2e}",
        f"{basic_metrics['reciprocity']:.1%}",
        f"{degree_stats['out_degree']['mean']:.1f}",
    ],
}

stats_df = pd.DataFrame(stats_data)
display(HTML(stats_df.to_html(index=False)))

Basic graph statistics

Metric	Value
Total Nodes	850,742
Total Edges	114,295,148
Source Nodes	850,263
Graph Density	1.58e-04
Reciprocity	70.7%
Average Degree	134.6

This is an extremely sparse network, typical of real-world graphs.

Degree Distributions

Out-Degree Distribution

The out-degree represents how many similar artists each node references.

out_degrees = np.array(distributions["out_degrees"])

# Create figure with histogram and KDE
fig = go.Figure()

# Histogram
fig.add_trace(
    go.Histogram(
        x=out_degrees,
        nbinsx=50,
        name="Histogram",
        opacity=0.5,
        histnorm="probability density",
        marker_color="lightseagreen",
    ),
)

# KDE
kde = gaussian_kde(out_degrees, bw_method=0.1)
x_range = np.linspace(out_degrees.min(), out_degrees.max(), 500)
kde_values = kde(x_range)

fig.add_trace(
    go.Scatter(
        x=x_range,
        y=kde_values,
        mode="lines",
        name="KDE",
        line=dict(color="darksalmon", width=3),
    ),
)

fig.update_layout(
    xaxis_title="Out-degree",
    yaxis_title="Density",
    height=500,
    showlegend=False,
    yaxis=dict(range=[0, None]),
    margin=dict(t=30),
)


# Add statistics
stats_text = f"Mean: {degree_stats['out_degree']['mean']:.1f}<br>"
stats_text += f"Median: {degree_stats['out_degree']['median']:.0f}<br>"
stats_text += f"Min: {degree_stats['out_degree']['min']:.0f}<br>"
stats_text += f"Max: {degree_stats['out_degree']['max']:,}<br>"
stats_text += f"Gini: {degree_stats['out_degree']['gini']:.3f}"

fig.add_annotation(
    text=stats_text,
    xref="paper",
    yref="paper",
    x=1.1,
    y=1,
    showarrow=False,
    font=dict(size=12),
    align="right",
    xanchor="right",
    yanchor="top",
)

fig.show()

In-Degree Distribution

The in-degree shows how many times each artist is referenced as similar by others.

in_degrees = np.array(distributions["in_degrees"])

# Create figure
fig = go.Figure()

# Histogram
fig.add_trace(
    go.Histogram(
        x=in_degrees,
        nbinsx=100,
        name="Histogram",
        opacity=0.5,
        histnorm="probability density",
        marker_color="plum",
    ),
)

# KDE
kde = gaussian_kde(in_degrees, bw_method=0.1)
x_range = np.linspace(in_degrees.min(), in_degrees.max(), 500)
kde_values = kde(x_range)

fig.add_trace(
    go.Scatter(
        x=x_range,
        y=kde_values,
        mode="lines",
        name="KDE",
        line=dict(color="darksalmon", width=3),
    ),
)

fig.update_layout(
    xaxis_title="In-degree",
    yaxis_title="Density",
    height=500,
    showlegend=False,
    yaxis=dict(range=[0, None]),
    margin=dict(t=30),
)

# Add statistics
stats_text = f"Mean: {degree_stats['in_degree']['mean']:.1f}<br>"
stats_text += f"Median: {degree_stats['in_degree']['median']:.0f}<br>"
stats_text += f"Min: {degree_stats['in_degree']['min']:,}<br>"
stats_text += f"Max: {degree_stats['in_degree']['max']:,}<br>"
stats_text += f"Gini: {degree_stats['in_degree']['gini']:.3f}"

fig.add_annotation(
    text=stats_text,
    xref="paper",
    yref="paper",
    x=1.1,
    y=1,
    showarrow=False,
    font=dict(size=12),
    align="right",
    xanchor="right",
    yanchor="top",
)

fig.show()

Power Law Analysis

Scale-free networks follow a power law distribution: P(k) ~ k^(-α)

# Create subplots
fig = make_subplots(
    rows=1,
    cols=2,
    subplot_titles=("Out-Degree Power Law", "In-Degree Power Law"),
)

# Out-degree power law
if "out_degree_fit" in power_law_fits:
    fit = power_law_fits["out_degree_fit"]

    # Calculate degree distribution from samples
    out_degrees = np.array(distributions["out_degrees"])
    unique_out, counts_out = np.unique(out_degrees[out_degrees > 0], return_counts=True)

    # Limit points for visualization
    if len(unique_out) > 100:
        indices = np.linspace(0, len(unique_out) - 1, 100, dtype=int)
        unique_out = unique_out[indices]
        counts_out = counts_out[indices]

    # Scatter plot of actual data
    fig.add_trace(
        go.Scatter(
            x=unique_out,
            y=counts_out,
            mode="markers",
            name="Actual",
            marker=dict(color="lightseagreen", size=8, opacity=0.5),
        ),
        row=1,
        col=1,
    )

    # Fitted line using the fit parameters
    x_fit = np.logspace(np.log10(fit["fit_range"][0]), np.log10(fit["fit_range"][1]), 100)
    y_fit = 10 ** (fit["intercept"]) * x_fit ** (-fit["alpha"])

    fig.add_trace(
        go.Scatter(
            x=x_fit,
            y=y_fit,
            mode="lines",
            name=f"Fit (α={fit['alpha']:.3f})",
            line=dict(color="darksalmon", width=3, dash="dash"),
        ),
        row=1,
        col=1,
    )

    # Add annotations
    annotation_text = f"R² = {fit['r_squared']:.4f}<br>"
    annotation_text += f"α = {fit['alpha']:.4f}"

    fig.add_annotation(
        text=annotation_text,
        xref="x domain",
        yref="y domain",
        x=0.3,
        y=0.98,
        bgcolor="#fafafa",
        showarrow=False,
        xanchor="right",
        yanchor="top",
    )

# In-degree power law
if "in_degree_fit" in power_law_fits:
    fit = power_law_fits["in_degree_fit"]

    # Calculate degree distribution from samples
    in_degrees = np.array(distributions["in_degrees"])
    unique_in, counts_in = np.unique(in_degrees[in_degrees > 0], return_counts=True)

    # Limit points for visualization
    if len(unique_in) > 100:
        indices = np.linspace(0, len(unique_in) - 1, 100, dtype=int)
        unique_in = unique_in[indices]
        counts_in = counts_in[indices]

    # Scatter plot
    fig.add_trace(
        go.Scatter(
            x=unique_in,
            y=counts_in,
            mode="markers",
            name="Actual",
            marker=dict(color="plum", size=8, opacity=0.5),
            showlegend=False,
        ),
        row=1,
        col=2,
    )

    # Fitted line using the fit parameters
    x_fit = np.logspace(np.log10(fit["fit_range"][0]), np.log10(fit["fit_range"][1]), 100)
    y_fit = 10 ** (fit["intercept"]) * x_fit ** (-fit["alpha"])

    fig.add_trace(
        go.Scatter(
            x=x_fit,
            y=y_fit,
            mode="lines",
            name=f"Fit (α={fit['alpha']:.3f})",
            line=dict(color="darksalmon", width=3, dash="dash"),
            showlegend=False,
        ),
        row=1,
        col=2,
    )

    # Add annotations
    annotation_text = f"R² = {fit['r_squared']:.4f}<br>"
    annotation_text += f"α = {fit['alpha']:.4f}"

    fig.add_annotation(
        text=annotation_text,
        xref="x2 domain",
        yref="y2 domain",
        x=0.3,
        y=0.98,
        showarrow=False,
        bgcolor="#fafafa",
        xanchor="right",
        yanchor="top",
    )

# Update axes to log scale
fig.update_xaxes(type="log", title="Degree (k)", row=1, col=1)
fig.update_xaxes(type="log", title="Degree (k)", row=1, col=2)
fig.update_yaxes(type="log", title="Frequency P(k)", row=1, col=1)
fig.update_yaxes(type="log", title="Frequency P(k)", row=1, col=2)

fig.update_layout(height=500, showlegend=False, margin=dict(t=40))

fig.show()

Edge Weight Distribution

weights = np.array(distributions["weights"])

# Create figure
fig = go.Figure()

# Histogram
fig.add_trace(
    go.Histogram(
        x=weights,
        nbinsx=100,
        name="Histogram",
        opacity=0.5,
        histnorm="probability density",
        marker_color="skyblue",
    ),
)

# KDE
kde = gaussian_kde(weights, bw_method=0.1)
x_range = np.linspace(0, 1, 500)
kde_values = kde(x_range)

fig.add_trace(
    go.Scatter(
        x=x_range,
        y=kde_values,
        mode="lines",
        name="KDE",
        line=dict(color="darksalmon", width=3),
    ),
)

fig.update_layout(
    xaxis_title="Similarity Score (1 - Weight)",
    yaxis_title="Density",
    height=500,
    showlegend=False,
    yaxis=dict(range=[0, None]),
    margin=dict(t=30),
)

# Add statistics
stats_text = f"Mean: {weight_stats['mean']:.3f}<br>"
stats_text += f"Median: {weight_stats['median']:.3f}<br>"
stats_text += f"Std: {weight_stats['std']:.3f}<br>"
stats_text += f"Min: {weight_stats['min']:.4f}<br>"
stats_text += f"Max: {weight_stats['max']:.4f}<br>"


fig.add_annotation(
    text=stats_text,
    xref="paper",
    yref="paper",
    x=1.1,
    y=1,
    showarrow=False,
    font=dict(size=12),
    align="right",
    xanchor="right",
    yanchor="top",
)

fig.show()

Top Connected Artists

# Create DataFrame for top nodes
top_in = pd.DataFrame(
    metrics["top_nodes"]["top_by_in_degree"][1:11],
    columns=["Artist", "In-Degree"],
)
top_out = pd.DataFrame(
    metrics["top_nodes"]["top_by_out_degree"][:10],
    columns=["Artist", "Out-Degree"],
)

# Create side-by-side display
html = f"""
<div style="display: flex; justify-content: space-around;">
    <div style="width: 45%;">
        <p style="font-size: .9rem; color: #5a6570; padding-top: .5rem; margin-bottom: -.2rem;">Top by In-Degree (Most Referenced)</p>
        {top_in.to_html(index=False)}
    </div>
    <div style="width: 45%;">
        <p style="font-size: .9rem; color: #5a6570; padding-top: .5rem; margin-bottom: -.2rem;">Top by Out-Degree (Most Connections Listed)</p>
        {top_out.to_html(index=False)}
    </div>
</div>
"""

display(HTML(html))

Top by In-Degree (Most Referenced)

Artist	In-Degree
Maetrik	2994
Lazy Hammock	2937
86 Mentality	2854
Don Gorda Project	2831
Mikko Alatalo	2817
Vargsang	2801
Rene Hell	2746
Pepe Willberg	2730
Epheles	2714
Noise Boyz	2689

Top by Out-Degree (Most Connections Listed)

Artist	Out-Degree
Olivia Rodrigo	250
Taylor Swift	250
Canaan Cox	249
Canaan Cox	249
Master	249
Morgan Evans	249
Morgan Evans	249
Hannah Ellis	248
Hannah Ellis	248
Massacra	248

Key Insights

Network Type

• Scale-free network with power-law in-degree distribution (α = 2.17)
• High reciprocity (70.7%) indicating symmetric similarity relationships
• Extremely sparse (density = 1.58e-04) yet well-connected

Degree Patterns

• Out-degree: Artificially uniform distribution due to API limits (max 250)
• In-degree: Natural power-law distribution with high inequality (Gini = 0.671)
• A few “supernode” artists are referenced by thousands of others

Weight Distribution

• Heavily skewed toward weak connections (median = 0.077)
• Most similarities are weak, suggesting the importance of thresholding in applications

Methodology

Data collected from Last.fm API for 850,742 artists. Analysis performed using streaming algorithms to handle the large graph size with memory constraints. Distributions sampled at 10% for weight analysis.